Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
1191 conjugacy classes of subgroups; only the 324 normal subgroups are shown.
This group has 1191 conjugacy classes of subgroups — too many to draw. Only its 324 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #120 |
| commutator | C23 | C24×C9 |
| Frattini, cosocle | C23×C3 | C24×C3 |
| socle | C22×C3 | C3×#120 |
| prime radical | C2×C3×H(5,𝔽2) | C2×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C24×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C9×#123 | C2 | semidirect |
| 3. C9×#458 | C2 | direct (N × H) |
| 4. C2×C9×#99 | C2 | semidirect |
| 11. C23×C4×C9 | C22 | semidirect |
| 13. C9×#123 | C22 | semidirect |
| 16. C2×C9×C22⋊C4 | C22 | semidirect |
| 17. C9×#99 | C22 | semidirect |
| 40. C22×C4×C9 | C23 | semidirect |
| 44. C9×C22⋊C4 | C23 | semidirect |
| 45. C2×#458 | C9 | direct (N × H) |
| 82. C2×#123 | C2×C9 | semidirect |
| 83. #458 | C2×C9 | direct (N × H) |
| 84. C2×#99 | C2×C9 | semidirect |
| 127. C23×C4 | C22×C9 | semidirect |
| 129. #123 | C22×C9 | semidirect |
| 132. C2×C22⋊C4 | C22×C9 | semidirect |
| 133. #99 | C22×C9 | semidirect |
| 160. C2×C9 | #458 | direct (N × H) |
| 180. C22×C4 | C23×C9 | semidirect |
| 184. C22⋊C4 | C23×C9 | semidirect |
| 199. C9 | C2×#458 | direct (N × H) |
| 246. C2 | C9×#458 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×H(5,𝔽2) | 2 |
| C2×C9×#123 | 2 |
| C9×#458 | 2 |
| C2×C9×#99 | 2 |
| C2×C3×#458 | 3 |